1. Part I: Basic Calculus, Real and Complex numbers Attempt four out of five problems 1. Explain what the real number system is? What is a typical real number? How are the sum and product of two typical real numbers defined? 2. Find all complex solutions of the equation 2(z + 1)3 = z 3 + (z 1)3 . Which one is closest to the imaginary unit i. 3. Find the following limits or show that they do not exisT: (a) lim x0+ (cos x) 1 x2 . (b) lim x (1 + 2 x 2 ) 2x . (c) limx0 4 + 3x 2 x . 4. Consider the function f(x) from the real numbers to the real numbers given by f(0) = 0 and f(x) = x 2 sin 1 x , when x 6= 0. Show that f 0 (0) exists and calculate it. Prove that the function f 0 (x) is not everywhere continuous.